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2015, 2015(special): 1060-1069. doi: 10.3934/proc.2015.1060

Simulation of complex dynamics using POD 'on the fly' and residual estimates

1. 

Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, 28911 Leganés, Spain

2. 

E.T.S.I. Aeronáutica y del Espacio, Universidad Politécnica de Madrid, 28040 Madrid, Spain

Received  September 2014 Revised  February 2015 Published  November 2015

Proper orthogonal decomposition (POD) is a very effective means to identify dynamical information contained in sets of snapshots that cover numerically computed trajectories of dissipative systems of partial differential equations. Such information is organized in a hierarchy of POD modes and the system can be Galerkin-projected onto the associated linear subspace. Quite frequently, the outcome is a low dimensional model of the problem. Flexibility and efficiency of the approximation can be enhanced if POD is applied `on the fly', adaptively combining a standard numerical solver (which provides the necessary snapshots) with the reduced system in interspersed intervals, as both time and a bifurcation parameter are varied. Residual estimates are introduced to make this adaptation accurate and robust, preventing possible mode truncation instabilities in the presence of complex dynamics. All ideas are illustrated in some bifurcation scenarios including quasi-periodic and chaotic attractors, which highlights a good computational efficiency.
Citation: Filippo Terragni, José M. Vega. Simulation of complex dynamics using POD 'on the fly' and residual estimates. Conference Publications, 2015, 2015 (special) : 1060-1069. doi: 10.3934/proc.2015.1060
References:
[1]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods,, Classics Appl. Math. 45, (2003). Google Scholar

[2]

D. Alonso, J. M. Vega and A. Velazquez, Reduced order model for viscous aerodynamic flow past an airfoil,, AIAA J., 48 (2010), 1946. Google Scholar

[3]

D. Alonso, J. M. Vega, A. Velazquez and V. de Pablo, Reduced-order modeling of three-dimensional external aerodynamic flows,, J. Aerospace Engrg., 25 (2012), 588. Google Scholar

[4]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Rev. Mod. Phys., 74 (2002), 100. Google Scholar

[5]

P. Astrid, S. Weiland, K. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition,, IEEE Trans. Automatic Control, 53 (2008), 2237. Google Scholar

[6]

M. J. Balajewicz, E. H. Dowell and B. R. Noack, Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation,, J. Fluid Mech., 729 (2013), 285. Google Scholar

[7]

M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models,, J. Comput. Phys., 228 (2009), 516. Google Scholar

[8]

T. Braconnier, M. Ferrier, J. C. Jouhaud, M. Montagnac and P. Sagaut, Towards an adaptive POD/SVD surrogate model for aeronautic design,, Computers & Fluids, 40 (2011), 195. Google Scholar

[9]

M. Couplet, C. Basdevant and P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modelling,, J. Comput. Phys., 207 (2005), 192. Google Scholar

[10]

M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems,, Cambridge University Press, (2009). Google Scholar

[11]

E. H. Dowell and K. C. Hall, Modeling of fluid-structure interaction,, Annu. Rev. Fluid Mech., 33 (2001), 445. Google Scholar

[12]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations,, J. Diff. Eq., 73 (1988), 309. Google Scholar

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications,, SIAM, (1977). Google Scholar

[14]

M. A. Grepl and A. T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations,, ESAIM: M2AN, 39 (2005), 157. Google Scholar

[15]

H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Bénard),, Comput. Meth. Appl. Mech. Engrg., 261-262 (2013), 261. Google Scholar

[16]

J. Heyman, G. Girault, Y. Guevel, C. Allery, A. Hamdouni and J. M. Cadou, Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method,, Computers & Fluids, 76 (2013), 73. Google Scholar

[17]

P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (1996). Google Scholar

[18]

M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation,, SIAM J. Appl. Dyn. Sys., 9 (2010), 1284. Google Scholar

[19]

Y. A. Kutnetsov, Elements of Applied Bifurcation Theory,, Appl. Math. Sci. 112, (2004). Google Scholar

[20]

T. Lieu, C. Farhat and M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration,, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 5730. Google Scholar

[21]

D. J. Lucia, P. S. Beran and W. A. Silva, Reduced-order modelling: new approaches for computations physics,, Prog. Aerosp. Sci., 40 (2004), 51. Google Scholar

[22]

M. Meis, F. Varas, A. Velazquez and J. M. Vega, Heat transfer enhancement in micro-channels caused by vortex promoters,, Int. J. Heat Mass Transfer, 53 (2010), 29. Google Scholar

[23]

M.-L. Rapún and J. M. Vega, Reduced order models based on local POD plus Galerkin projection,, J. Comput. Phys., 229 (2010), 3046. Google Scholar

[24]

M.-L. Rapún, F. Terragni and J. M. Vega, Adaptive POD-based low-dimensional modeling supported by residual estimates,, Int. J. Numer. Meth. Engng (2015), (2015). Google Scholar

[25]

D. Rempfer, On low-dimensional Galerkin models for fluid flow,, Theor. Comput. Fluid Dyn., 14 (2000), 75. Google Scholar

[26]

S. Sirisup and G. E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models,, J. Comput. Phys., 194 (2004), 92. Google Scholar

[27]

S. Sirisup, G. E. Karniadakis, D. Xiu and I. G. Kevrekidis, Equations-free/Galerkin-free POD assisted computation of incompressible flows,, J. Comput. Phys., 207 (2005), 568. Google Scholar

[28]

L. Sirovich, Turbulence and the dynamics of coherent structures,, Q. Appl. Math., XLV (1987), 561. Google Scholar

[29]

F. Terragni, E. Valero and J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem,, SIAM J. Sci. Comput., 33 (2011), 3538. Google Scholar

[30]

F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems,, Physica D, 241 (2012), 1393. Google Scholar

[31]

F. Terragni and J. M. Vega, Construction of bifurcation diagrams using POD on the fly,, SIAM J. Appl. Dyn. Syst., 13 (2014), 339. Google Scholar

show all references

References:
[1]

E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods,, Classics Appl. Math. 45, (2003). Google Scholar

[2]

D. Alonso, J. M. Vega and A. Velazquez, Reduced order model for viscous aerodynamic flow past an airfoil,, AIAA J., 48 (2010), 1946. Google Scholar

[3]

D. Alonso, J. M. Vega, A. Velazquez and V. de Pablo, Reduced-order modeling of three-dimensional external aerodynamic flows,, J. Aerospace Engrg., 25 (2012), 588. Google Scholar

[4]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Rev. Mod. Phys., 74 (2002), 100. Google Scholar

[5]

P. Astrid, S. Weiland, K. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition,, IEEE Trans. Automatic Control, 53 (2008), 2237. Google Scholar

[6]

M. J. Balajewicz, E. H. Dowell and B. R. Noack, Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier-Stokes equation,, J. Fluid Mech., 729 (2013), 285. Google Scholar

[7]

M. Bergmann, C. H. Bruneau and A. Iollo, Enablers for robust POD models,, J. Comput. Phys., 228 (2009), 516. Google Scholar

[8]

T. Braconnier, M. Ferrier, J. C. Jouhaud, M. Montagnac and P. Sagaut, Towards an adaptive POD/SVD surrogate model for aeronautic design,, Computers & Fluids, 40 (2011), 195. Google Scholar

[9]

M. Couplet, C. Basdevant and P. Sagaut, Calibrated reduced-order POD-Galerkin system for fluid flow modelling,, J. Comput. Phys., 207 (2005), 192. Google Scholar

[10]

M. Cross and H. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems,, Cambridge University Press, (2009). Google Scholar

[11]

E. H. Dowell and K. C. Hall, Modeling of fluid-structure interaction,, Annu. Rev. Fluid Mech., 33 (2001), 445. Google Scholar

[12]

C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolution equations,, J. Diff. Eq., 73 (1988), 309. Google Scholar

[13]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications,, SIAM, (1977). Google Scholar

[14]

M. A. Grepl and A. T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations,, ESAIM: M2AN, 39 (2005), 157. Google Scholar

[15]

H. Herrero, Y. Maday and F. Pla, RB (Reduced basis) for RB (Rayleigh-Bénard),, Comput. Meth. Appl. Mech. Engrg., 261-262 (2013), 261. Google Scholar

[16]

J. Heyman, G. Girault, Y. Guevel, C. Allery, A. Hamdouni and J. M. Cadou, Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method,, Computers & Fluids, 76 (2013), 73. Google Scholar

[17]

P. Holmes, J. L. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge University Press, (1996). Google Scholar

[18]

M. Ilak, S. Bagheri, L. Brandt, C. W. Rowley and D. S. Henningson, Model reduction of the nonlinear complex Ginzburg-Landau equation,, SIAM J. Appl. Dyn. Sys., 9 (2010), 1284. Google Scholar

[19]

Y. A. Kutnetsov, Elements of Applied Bifurcation Theory,, Appl. Math. Sci. 112, (2004). Google Scholar

[20]

T. Lieu, C. Farhat and M. Lesoinne, Reduced-order fluid/structure modeling of a complete aircraft configuration,, Comput. Meth. Appl. Mech. Engrg., 195 (2006), 5730. Google Scholar

[21]

D. J. Lucia, P. S. Beran and W. A. Silva, Reduced-order modelling: new approaches for computations physics,, Prog. Aerosp. Sci., 40 (2004), 51. Google Scholar

[22]

M. Meis, F. Varas, A. Velazquez and J. M. Vega, Heat transfer enhancement in micro-channels caused by vortex promoters,, Int. J. Heat Mass Transfer, 53 (2010), 29. Google Scholar

[23]

M.-L. Rapún and J. M. Vega, Reduced order models based on local POD plus Galerkin projection,, J. Comput. Phys., 229 (2010), 3046. Google Scholar

[24]

M.-L. Rapún, F. Terragni and J. M. Vega, Adaptive POD-based low-dimensional modeling supported by residual estimates,, Int. J. Numer. Meth. Engng (2015), (2015). Google Scholar

[25]

D. Rempfer, On low-dimensional Galerkin models for fluid flow,, Theor. Comput. Fluid Dyn., 14 (2000), 75. Google Scholar

[26]

S. Sirisup and G. E. Karniadakis, A spectral viscosity method for correcting the long-term behavior of POD models,, J. Comput. Phys., 194 (2004), 92. Google Scholar

[27]

S. Sirisup, G. E. Karniadakis, D. Xiu and I. G. Kevrekidis, Equations-free/Galerkin-free POD assisted computation of incompressible flows,, J. Comput. Phys., 207 (2005), 568. Google Scholar

[28]

L. Sirovich, Turbulence and the dynamics of coherent structures,, Q. Appl. Math., XLV (1987), 561. Google Scholar

[29]

F. Terragni, E. Valero and J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem,, SIAM J. Sci. Comput., 33 (2011), 3538. Google Scholar

[30]

F. Terragni and J. M. Vega, On the use of POD-based ROMs to analyze bifurcations in some dissipative systems,, Physica D, 241 (2012), 1393. Google Scholar

[31]

F. Terragni and J. M. Vega, Construction of bifurcation diagrams using POD on the fly,, SIAM J. Appl. Dyn. Syst., 13 (2014), 339. Google Scholar

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